Showing the polynomial $x^4 + x^3 + 4x + 1$ is irreducible in $\mathbb{Q}[x]$.

Question

Showing the polynomial $f(x) = x^4 + x^3 + 4x + 1$ is irreducible in $\mathbb{Q}[x]$.

I have two question relating to this which I've bolded below.

Attempt to answer

(*) $f$ has degree $4 \ge 2$ so if $f$ has a root then $f$ is reducible.

Well the rational root test says that if $\exists$ a rational root $\frac{p}{q}$ with $p, q$ coprime then $p|a_0$ and $q|a_n$.

Hence the potential rational roots are $\pm 1$. Neither of these are roots so $f$ does not have a root. But (*) only says that if $f$ has a root then $f$ is reducible, it does not imply that if $f$ does not have a root then $f$ is irreducible. Is this correct?

Now we can't apply Eisenstein's irreducibility criterion theorem as there is no prime $p$ such that

$p\mid a_0, a_1,..., a_{n-1}$

$p \nmid a_n$

$p^2 \nmid a_0$

So where do we go from here? One other fact I am aware of is that if $f$ is primitive, which it is, $f$ irreducible in $Q[x] \iff f$ is irreducible in $Z[x]$. But I don't see how that can help me here. So how do I proceed from here?

Answer 1

it does not imply that if f does not have a root then f is irreducible. Is this correct?

Yes, that is correct. For polynomials of degree $> 3$, the absence of roots in $\mathbb{Q}$ is only a necessary, but not a sufficient condition for irreducibility.

Aside from trying a substitution $x \mapsto x - a$ to achieve a form where Eisenstein's criterion is applicable, you can consider the corresponding polynomial over $\mathbb{Z}/(p)$ for a prime $p$. If $f$ is reducible, so is its image $\overline{f} \in \left(\mathbb{Z}/(p)\right)[X]$, if it has the same degree as $f$ (since $f$ here is monic, that is the case). So if $\overline{f}$ is irreducible in any $\left(\mathbb{Z}/(p)\right)[X]$, then $f$ itself is irreducible.

Note: if $\overline{f}\in \left(\mathbb{Z}/(p)\right)[X]$ is reducible, that does not imply that $f$ is reducible.

If we look at the example over $\mathbb{Z}/(2)$, we have $\overline{f}(x) = x^4+x^3+1$. It is readily seen that that has no zero in $\mathbb{Z}/(2)$, so if it were reducible, both factors would have to be quadratic. Since the constant term is $1$, a factorisation would have the form $$x^4+x^3+1 = (x^2+ax+1)(x^2+bx+1).$$ But then the coefficients of $x^3$ and $x^1$ would be equal ($a+b$). So $\overline{f}$ is irreducible.

Answer 2

Another way, instead of trying to apply Eisenstein's, is to assume that you can indeed factor it $(x^2+bx+c)(x^2+dx+e)=x^4+x^3+4x+1$ and arrive at a contradiction. Gauss' lemma makes your work a lot easier too.

Answer 3

Apply Eisenstein's criterion for $p=3$ to $$f(x-1)=x^4-3\,x^3+3\,x^2+3\,x-3$$

If $f$ can be factored, both factors must have degree $2$ becuase there is no integer root. Because of $$f(-11)=13267,f(-3)=43, f(-2)=1, f(-1)=-3, f(0)=1, f(1)=7,f(4)=337$$ one of these degree $2$-factors is $1$ at three x-values (or -1). So it is identical to $1$ (or $-1$). So $f$ is irreducible. See this link ($13267,43,3,7,337$ are primes)

But it is sufficient that $$f(-11)=13267$$ is a prime because $11 \gt 2+\max\{\frac{|a_i|}{a_4} \mid i =0,1,2,3\}$ See this and this

Answer 4

Hint: try a substitution $x \mapsto x - a$ for some $a$; if the resulting polynomial is irreducible (using Eisenstein, for examples), your original one is as well.

Answer 5

Both your questions were answered by Daniel Fischer.

Here is another way to prove the irreducibility of that polynomial.

As you noted the polynomial $f$ is irreducible in $\mathbb Q[x] \iff$ is irreducible in $\mathbb Z[x]$. Note also that $f$ has only one root inside the unit circle and all other roots outside the unit circle (use Rouche's theorem).
Now if we suppose that $f$ is reducible with $f(x)=g(x)h(x)$ (where $g(x),h(x)\in\mathbb Z[x], \ \deg f,\deg g>0$) then one of the polynomials $g,h$ has all of its roots outside the unit circle and constant term $\pm1$ (which is the product of its roots). This is a contradiction.